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Chapter Learning Outcomes After completing this chapter, the student will be able to: Find a mathematical model, called a state-space representation, for a linear, time invariant system (Sections ) Model electrical and mechanical systems in state space (Section 3.4) Convert a transfer function to state space (Section 3.5) Convert a state-space representation to a transfer function (Section 3.6) Linearize a state-space representation (Section 3.7) Basil Hamed2

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3.1 Introduction Two approaches are available for the analysis and design of feedback control systems. The first, which we began to study in Chapter 2, is known as the classical, or frequency-domain, technique. The 1 st approach is based on converting a system's differential equation to a transfer function, thus generating a mathematical model of the system that algebraically relates a representation of the output to a representation of the input. The primary disadvantage of the classical approach is its limited applicability: It can be applied only to linear, time- invariant systems or systems that can be approximated as such. Basil Hamed4

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3.1 Introduction The 2nd approach is state-space approach (also referred to as the modern, or time-domain, approach) is a unified method for modeling, analyzing, and designing a wide range of systems. For example, the state-space approach can be used to represent nonlinear systems, Time-varying systems, Multiple- input, multiple-output systems. The time-domain approach can also be used for the same class of systems modeled by the classical approach. Basil Hamed5

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3.2 Some Observations We proceed now to establish the state-space approach as an alternate method for representing physical systems. In general, an nth-order differential equation can be decomposed into n first-order differential equations. Because, in principle, first-order differential equations are simpler to solve than higher-order ones, first-order differential equations are used in the analytical studies of control systems. Basil Hamed6

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3.2 Some Observations Definition of State Variables The state of a system refers to the past, present, and future conditions of the system. From a mathematical perspective, it is convenient to define a set of state variables and state equations to model dynamic systems. As it turns out, the variables x 1 (t), x 2 (t),...,x„(t) are the state variables of the nth-order system Basil Hamed7

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3.3The General State-Space Representation Where Basil Hamed10 The state variables of a system are defined as a minimal set of variables, x1(t),x2(t),...,xn(t), such that knowledge of these variables at any time to and information on the applied input at time t0 are sufficient to determine the state of the system at any time t > to

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Example Basil Hamed11 Given 2 nd order Diff Eq. Above eq. can be transform into state eq; Let then Eq. (1) is decomposed into the following two first-order differential equations: 1

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Example Basil Hamed12

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General form of state Space model Basil Hamed13 In general, the differential equation of an nth-order system is written let us define then the nth-order differential equation is decomposed into n first-order differential equations:

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3.4 Applying the State-Space Representation In this section, we apply the state-space formulation to the representation of more complicated physical systems. The first step in representing a system is to select the state vector, which must be chosen according to the following considerations: 1. A minimum number of state variables must be selected as components of the state vector. This minimum number of state variables is sufficient to describe completely the state of the system. 2. The components of the state vector (that is, this minimum number of state variables) must be linearly independent. Basil Hamed14

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Linearly Independent State Variables The components of the state vector must be linearly independent. For example, following the definition of linear independence, if x1, x2, and x3 are chosen as state variables, but x3 = 5x1 + 4x2, then x3 is not linearly independent of x1and x2, since knowledge of the values of x1 and x2 will yield the value of x3. Basil Hamed15

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Minimum Number of State Variables Typically, the minimum number required equals the order of the differential equation describing the system. For example, if a third-order differential equation describes the system, then three simultaneous, first-order differential equations are required along with three state variables. From the perspective of the transfer function, the order of the differential equation is the order of the denominator of the transfer function after canceling common factors in the numerator and denominator. Basil Hamed 16

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Minimum Number of State Variables In most cases, another way to determine the number of state variables is to count the number of independent energy- storage elements in the system. The number of these energy-storage elements equals the order of the differential equation and the number of state variables. Basil Hamed17

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Example Basil Hamed18 Find state model of System shown in the Fig. Solution A practical approach is to assign the current in the inductor L, i(t), and the voltage across the capacitor C, ec(t), as the state variables. The reason for this choice is because the state variables are directly related to the energy-storage element of a system. The inductor stores kinetic energy, and the capacitor stores electric potential energy. By assigning i(t) and ec(t) as state variables, we have a complete description of the past history (via the initial states) and the present and future states of the network.

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Example The state equation : Basil Hamed19 This format is also known as the state form if we set OR

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Example Basil Hamed 20 write the state equations of the electric network shown in the Fig. Solution: The state equations of the network are obtained by writing the voltages across the inductors and the currents in the capacitor in terms of the three state variables. The state equations are

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Example In vector-matrix form, the state equations are written as Basil Hamed21 Where

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Example 3.1 P.138 PROBLEM: Given the electrical network of Figure shown, find a state-space representation if the output is the current through the resistor. Basil Hamed22 Solution Select the state variables by writing the derivative equation for all energy storage elements, that is, the inductor and the capacitor. Thus, 1212

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Example 3.1 Apply network theory, such as Kirchhoffs voltage and current laws, to obtain ic and vL in terms of the state variables, v c and i L. At Node 1, Basil Hamed23 which yields ic in terms of the state variables, vc and i L. Around the outer loop, 3 4

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Example 3.1 Substitute the results of Eqs. (3) and (4) into Eqs. (1) and (2) to obtain the following state equations: Basil Hamed 24 OR Find the output eq. since the output is i R (t) The final result for the state-space representation is

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Example Basil Hamed25 Find the state eq. of the mechanical system shown Solution

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Example 3.3 P.142 PROBLEM: Find the state equations for the translational mechanical system shown in Figure. Basil Hamed26

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Example 3.3 P.142 SOLUTION: First write the differential equations for the network in Figure, using the methods of Chapter 2 to find the Laplace-transformed equations of motion. Basil Hamed27

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Example 3.3 P.142 Basil Hamed28 In Vector Matrix

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3.5 Converting a Transfer Function to State Space In the last section, we applied the state-space representation to electrical and mechanical systems. We learn how to convert a transfer function representation to a state-space representation in this section. One advantage of the state-space representation is that it can be used for the simulation of physical systems on the digital computer. Thus, if we want to simulate a system that is represented by a transfer function, we must first convert the transfer function representation to state space. Basil Hamed29

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Converting T.F to S.S System modeling in state space can take on many representations Although each of these models yields the same output for a given input, an engineer may prefer a particular one for several reasons. Another motive for choosing a particular set of state variables and state-space model is ease of solution. Basil Hamed30

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Converting T.F to S.S There are many ways of converting T.F into S.S but the most useful and famous are: 1. Direct Decomposition 2. Cascade Decomposition 3. Parallel Decomposition Basil Hamed31

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Cascade (Series) Decomposition Now write the state equations for the new representation of the system. The state-space representation is completed by rewriting above Eqs in vector-matrix form: Basil Hamed40

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Parallel Decomposition Parallel subsystems have a common input and an output formed by the algebraic sum of the outputs from all of the subsystems. Basil Hamed 41

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Parallel Decomposition Example Basil Hamed42

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Parallel Decomposition Basil Hamed43

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Parallel Decomposition Basil Hamed44

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3.6 Converting from State Space to a Transfer Function Basil Hamed45

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Example 3.6 Basil Hamed50 we obtain the final result for the transfer function:

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3.7 Linearization A prime advantage of the state-space representation over the transfer function representation is the ability to represent systems with nonlinearities. A linearized model is valid only for limited range of operation, and often only at the operating point at which the linearized is carried out. Basil Hamed51

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Example of Nonlinear Basil Hamed55 Because nonlinear systems are usually difficult to analyze and design, it is desirable to perform a linearization whenever the situation justifies it. A linearization process that depends on expanding the nonlinear state equations into a Taylor series about a nominal operating point or trajectory